Generators for comonoids and universal constructions

TL;DR

This paper explores the construction of cofree coalgebras and limits in various abelian monoidal categories, providing explicit generators and solutions to open questions about cofree corings and module coalgebras.

Contribution

It introduces concrete generators for coalgebra categories and explicitly constructs cofree coalgebras, addressing open problems in the theory.

Findings

Constructed cofree coalgebras in several categories

Provided explicit generators for coalgebra categories

Solved open problem on existence of cofree coring

Abstract

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question of A. Agore on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a $B$-(co)module, for a bialgebra $B$.